Prolog Programming Volume 2

1
Prolog Programming(Volume 2)

• Dr W.F. Clocksin

2
Lists

• Lists are the same as other languages (such as
  ML) in that a list of terms of any length is
  composed of list cells that are consed
  together.
• The list of length 0 is called nil, written .
• The list of length n is .(head,tail), where tail
  is a list of length n-1.
• So a list cell is a functor . of arity 2. Its
  first component is the head, and the second
  component is the tail.

3
Examples of lists

• nil
• .(a, nil)
• .(a, .(b, nil)
• .(a, .(b, .(c, .(d, .(e. nil)))))
• .(a,b) (note this is a pair, not a proper list)
• .(a, X) (this might be a list, or might not!)
• .(a, .(b, nil)), .(c, nil))

4
They can be written as trees
a
a
nil
b
nil
a
b
a
X
a
b
c
a
a
nil
d
b
nil
e
nil
5
Prolog Syntax for Lists

• Nil is written .
• The list consisting of n elements t1, t2, ,tn is
  written t1, t2, ,tn.
• .(X,Y) is written XY
• X is written X
• The term .(a, .(b, .(c,Y))) is written a,b,cY.
• If Y is instantiated to , then the term is a
  list, and can be written a,b,c or simply
  a,b,c.

6
Exercises

• Identify the heads and tails of these lists (if
  any)
• a, b, c
• a
• the, cat, sat
• the, cardinal, pulled, off, each, plum,
  coloured, shoe

7
Exercises

• Identify the heads and tails of these lists (if
  any)
• a b, c
• a
• the, cat, sat
• the, cardinal, pulled, off, each, plum,
  coloured, shoe

8
Exercises

• Identify the heads and tails of these lists (if
  any)
• a, b, c
• a
• the, cat, sat
• the, cardinal, pulled, off, each, plum,
  coloured, shoe

9
Exercises

• Identify the heads and tails of these lists (if
  any)
• a, b, c
• a
• (not a list, so doesnt have head and tail.
  nil is a constant)
• the, cat, sat
• the, cardinal, pulled, off, each, plum,
  coloured, shoe

10
Exercises

• Identify the heads and tails of these lists (if
  any)
• a, b, c
• a
• the, cat sat
• the, cardinal, pulled, off, each, plum,
  coloured, shoe

11
Exercises

• Identify the heads and tails of these lists (if
  any)
• a, b, c
• a
• the, cat, sat
• the, cardinal pulled, off, each, plum,
  coloured, shoe

12
Exercises

• For each pair of terms, determine whether they
  unify, and if so, to which terms are the
  variables instantiated?
• X, Y, Z john, likes, fish
• cat XY
• X,YZ mary, likes, wine (picture on next
  slide)
• the,YZ X,answer, is, here
• X, Y, X a, Z, Z
• X, Y, X a, X, X

13
Remember

• A variable may be instantiated to any term.

X
Y
Z
mary
likes
wine

mary, likes, wine X,YZ
14
Fun with Lists (Worksheet 5)

• / member(Term, List) /
• member(X, XT).
• member(X, HT) - member(X, T).
• Examples
• ?- member(john, paul, john).
• ?- member(X, paul, john).
• ?- member(joe, marx, darwin, freud).
• ?- member(foo, X).

15
Exercises

• Here is a mystery predicate. What does it do?
• mystery(X, A, B) - member(X, A), member(X, B).
• ?- mystery(a, b, c, a, p, a, l).
• ?- mystery(b, b, l, u, e, y, e, l, l, o, w).
• ?- mystery(X, r, a, p, i, d, a, c, t, i, o,
  n).
• ?- mystery(X, w, a, l, n, u, t, c, h, e, r, r,
  y).

16
A Brief Diversion intoAnonymous Variables

• / member(Term, List) /
• member(X, XT).
• member(X, HT) - member(X, T).
• member(X, X_).
• member(X, _T) - member(X, T).

Notice T isnt used
Notice H isnt used
17
A Brief Diversion into Arithmetic

• The built-in predicate is takes two arguments.
  It interprets its second as an arithmetic
  expression, and unifies it with the first. Also,
  is is an infix operator.
• ?- X is 2 2 2.
• X 6
• ?- 10 is (2 0) 2 ltlt 4.
• no
• ?- 32 is (2 0) 2 ltlt 4.
• yes

18
is

• But is cannot solve equations, so the
  expression must be ground (contain no free
  variables.
• ?- Y is 2 X.
• no
• ?- X is 15, Y is 2 X.
• X 15, Y 30.

19
Worksheet 6 Length of a List

• / length(List, Length) /
• Naïve method
• length(, 0).
• length(HT, N) - length(T, NT), N is NT 1.

20
Worksheet 6

• / length(List, Length) /
• Tail-recursive method
• length(L, N) - acc(L, 0, N).
• / acc(List, Before, After) /
• acc(, A, A).
• acc(HT, A, N) - A1 is A 1, acc(T, A1, N).

21
Exercises

• ?- length(apple, pear, N).
• ?- length(alpha, 2).
• ?- length(L, 3).
• Modify length to give a procedure sum such that
  sum(L,N) succeeds if L is a list of integers and
  N is their sum.

22
Worksheet 7 Inner Product

• A list of n integers can be used to represent an
  n-vector (a point in n-dimensional space). Given
  two vectors a and b, the inner product (dot
  product) of a and b is defined
• As you might expect, there are naïve and
  tail-recursive ways to compute this.

23
Worksheet 7

• The naïve method
• inner(, , 0).
• inner(AAs,BBs,N) -
• inner(As, Bs, Ns), N is Ns (A B).

24
Worksheet 7

• Tail-recursive method
• inner(A, B, N) - dotaux(A, B, 0, N).
• dotaux(, , V, V).
• dotaux(AAs,BBs,N,Z) -
• N1 is N (A B),
• dotaux(As, Bs, N1, Z).

25
Worksheet 8 Maximum of a List

• Tail-recursive method has a base case and two
  recursive cases
• / max(List, Accumulator, Result) /
• max(, A, A).
• max(HT, A, M) - H gt A, max(T, H, M).
• max(HT, A, M) - H lt A, max(T, A, M).
• How to initialise the accumulator?

26
Worksheet 8

• maximum(L, M) - max(L, -10000, M).
• Magic numbers are a bad idea.
• maximum(L, M) - max(L, minint, M)
• Need to change definitions of arithmetic.
• maximum(HT, M) - max(T, H, M).
• And know that ?- maximum(,X) fails.

27
Worksheet 9
arc
Here we have added a new arc from d to a. If you
use the program from WS 4, some goals will cause
an infinite loop.

• a(g, h).
• a(d, a).
• a(g, d).
• a(e, d).
• a(h, f).
• a(e, f).
• a(a, e).
• a(a, b).
• a(b, f).
• a(b, c).
• a(f, c).

a
b
c
e
d
f
g
h

Keep a trail of nodes visited so far. Visit
only legal nodes, not already on the trail.
Represent the trail as an accumulator, an extra
argument of path.
28
Worksheet 9

• path(X, X, T).
• path(X, Y, T) -
• a(X, Z), legal(Z, T), path(Z, Y, ZT).
• legal(Z, ).
• legal(Z, HT) - Z \ H, legal(Z, T).
• Notice that legal is like the negation of member.